Perfect Squares (original) (raw)
Last Updated : 8 Jan, 2026
An integer that can be written as the product of two equal integers is called a perfect square. For example, 36 is a perfect square because it is the product of 6 ⨉ 6. In other words, a perfect square is n raised to the power 2, where n is an integer.
A perfect square can also be visualized in real life through square-shaped objects like these
**Other examples of perfect square:
- 9 is a perfect square as we can write it as 3 x 3
- 16 is a perfect square as we can write it as 4 x 4
18 is NOT a perfect square as we cannot write as x2 for an integer x
**Properties of Perfect Squares
Perfect squares have unique characteristics that differentiate them from other numbers.
- Perfect Squares are the only numbers that have odd number of distinct divisors. For example, 9 has divisors as 1, 3, and 9. 4 has divisors as 1, 2 and 4. For all other numbers, divisors appear in pair, so the total number of divisors is always even for other numbers.
- A perfect square always ends in one of these digits: 0, 1, 4, 5, 6, or 9. It will never end in 2, 3, 7, or 8.
- Consecutive perfect squares increase by odd numbers pattern of differences is: 1, 3, 5, 7, 9, …
Please refer Perfect Square Interesting Facts for details.
Perfect Squares of 1 to 30
Chart for Perfect Square from 1 to 30 is added below as:
Perfect Square Formula
The formula for a perfect square is expressed as **n 2, where '**n' is a whole number. In this formula, n is multiplied by itself, resulting in a perfect square. For example, if n is 3, the perfect square is 32, which equals 9.
Other formulas used for perfect square are,
- **n 2 − (n − 1) 2 = 2n − 1
- **n 2 = (n − 1) 2 + (n − 1) + n
Algebraic Identities as perfect squares:
- **a 2 + 2ab + b 2 = (a + b) 2
- **a 2 – 2ab + b 2 = (a – b) 2
Tips and Tricks to Identify Perfect Square Numbers
Identifying perfect square numbers can be simplified with some handy tips and tricks. These methods can help you quickly determine whether a number is a perfect square, even without a calculator.
1. Check the Last Digit
**Perfect squares end in specific digits: Perfect square numbers always end in 0, 1, 4, 5, 6, or 9. If a number ends in 2, 3, 7, or 8, it cannot be a perfect square. This is a quick way to eliminate many numbers from consideration.
2. Find Perfect Squares by Adding Odd Numbers
**Sum of Odd Numbers: Addition of first n odd numbers is always perfect square. For example 1 + 3 = 4, 1 + 3 + 5 = 9,
1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 + 11 = 36 ...
This pattern shows that the sum of the first (n) odd numbers is always a perfect square equal to (n^2). This is a useful trick for quickly identifying perfect squares, especially when dealing with smaller numbers.
Check here the explanation and proof of above approach.
3. Approximate the Square Root
**Estimating the Square Root: You can estimate the square root of a number and check if the result is an integer. If the square root of a number is an integer, then the number is a perfect square. For example, the square root of 49 is 7, which confirms that 49 is a perfect square.
4. Identify the Number of Zeros
**Even Number of Zeros: If a number has an even number of zeros at the end, it might be a perfect square. For instance, 100 (which is (10^2)) and 400 (which is (20^2)) have even numbers of zeros and are perfect squares. A number like 1000, with an odd number of zeros, cannot be a perfect square.
5. Apply Factorization
**Prime Factorization: If a number's prime factors all have even exponents, then it is a perfect square. For example, the prime factorization of 36 is (2^2 times 3^2), where both exponents are even, confirming that 36 is a perfect square.
6. Use Special Number Patterns
**Square of a Number Ending in 5: To find square of a number ending in 5, multiply the digit before 5 with next digit and append 25. For example, 752= 7×8(25) = 5625
**Square of Numbers Close to 100: For numbers close to 100, express the square as (100 - x)2= 1002 - 200x + x2. This simplifies calculations, especially for mentally calculating squares.
**Odd Number Squares: Square of any odd number is an odd number. If n is an odd number, then n2 is odd.
**Even Number Squares: Square of any even number is an even number. If m is an even number, then m2 is even.
**Difference of Squares: Use difference of squares formula, a2− b2= (a+b)(a−b). This can help in factoring or simplifying expressions.
**Square of a Sum: (a+b)2 = a2 + 2ab + b2
**Square of a Difference: (a−b)2 = a2 − 2ab + b2
7. Consider Modulo Properties
**Modulo 4 and 3 Properties: A perfect square, when divided by 4, leaves a remainder of 0 or 1. Similarly, when divided by 3, a perfect square leaves a remainder of 0 or 1. These properties help to quickly determine whether a number could be a perfect square.
8. Use the Digital Root Method
**Digital Root of Perfect Squares: The digital root of a perfect square is always 1, 4, 7, or 9. The digital root is found by summing all the digits of the number repeatedly until a single-digit number is obtained. For example, the digital root of 81 is 9 (since (8 + 1 = 9)), which is one of the valid digital roots for a perfect square.
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**Perfect Square Solved Examples
**Example 1: Identify the first two perfect squares.
**Solution:
First two perfect squares are obtained by squaring the first two whole numbers:
- 12=1 (Square of 1 is 1)
- 22= 42 (Square of 2 is 4)
Therefore, first two perfect squares are 1 and 4.
**Example 2: If a number is a perfect square and its square root is 9, what is the number?
**Solution:
If a number is a perfect square and its square root is 9, we can find the number by squaring the square root:
92 = 81
So, required number is 81, as it is a perfect square, and its square root is 9.
**Example 3: If a number is a perfect square and its square root is a prime number, find the number.
**Solution:
Take the prime number 5. The square of 5 is 25 (52=25). Here, 25 is a perfect square, and 5 is a prime number.
So, the number we're looking for is 25, where the square root (5) is a prime number
Practice Questions on Perfect Square
Some questions on perfect square are:
**Question 1: Find the square of 5.
**Question 2: Is 36 a perfect square?
**Question 3:. Determine the square root of 49.
**Question 4: Write next two perfect squares after 16.
**Question 5: Identify the perfect square closest to 150.